Geometric structures and invariants of links and 3-manifolds

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The research of this project falls in the area of 3-dimensional topology. The central objects of study in this area are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of low dimensional topology. The main theme of this project is to establish such connections and explore their ramifications and applications to other areas of mathematics.This project will establish relationships between geometry and combinatorial descriptions, properties, and quantum invariants of links and 3-manifolds. The PI has developed a setting for establishing new unexpected relations between the colored Jones link polynomials, the topology and geometry of essential surfaces in link complements, and hyperbolic geometry. One part of the project will continue developing this theory and exploring its applications. Another part, will combine several new techniques, to develop methods for recognizing geometric structures on 3-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data. A third part will study skein link theory in 3-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds. A fourth part will explore the applicability of quantum knot invariants to classical questions in knot theory and search for a classification of crossing changes that do not alter the topology of the underlying knots. The project also involves the research of graduate students currently working with PI.